\section{American Option}
An American styled derivative is a contract that can be exercised during the
life of an option. The optimal strategy is the one, which will provide the
holder of the option with a maximum value. Ofcourse, one does not know ahead
of time what the optimal exercise strategy is. In practice, the optimal
exercise strategy is found simutaneously with the price. It is this optimal
exercise strategy that is associated with the concept of free boundaries. The
free boundary seperates the region where it is optimal to exercise the option
from the region where it is optimal to hold the option. The dividing price
between exercise and non-exercise is called the optimal exercise price. It
depends on the time remaining to experiry as well as the other parameters such
as the volatility. Unlike European style instruments a free/moving boundary is
formed because the optimal exercise price is not known a-priori as a funcion
of time.
\noindent
For a vanilla Amercian option the value of the option can be determined using
the `Bellman principle of dynamics programming', which states:\\
\\
\textit{
``At a given time the optimal strategy corresponds to the maximum either the
exercise value, or the value associated with selecting an optimal strategy an
instant later."}\\
\\
Or in formula form:\\
\[
V(S,t)=\max(f(S), PV_t[V(S+ds, t+dt)])
\]
where \\
\begin{tabular}{cl}
$S=S(t)$& underlying stock price process at time t\\
$f(S)$& exercise or intrinsic value dependent only on $S(t)$\\
$dS$& a small change in the stock price\\
$dt$& a small change in time\\
$PV_t$& present value at time $t$.\\
\end{tabular}
\\
For an European put option we have $V(S_T,T)=\max(K-S_T,0)$ for an American
put option we have $V(S_t,t)\geq\max(K-S_t,0)$ due to the possibility of early
exersize. This obviously leads to the two boundary conditions:
\begin{eqnarray*}
0&\leq& V(S_t,t)\\
K&\geq&V(S_t,t)
\end{eqnarray*}
The convexity of the put price (we assume the put price is convex) togethe
with the equations given above imply that
$\lim_{s\rightarrow\infty}\frac{dV}{dS_t}(S_t,t)=0$. If we rewrite this again
to finite difference this results in $V_{N}^{\tau}-V_{N+1}^{\tau}=0$ $\forall
\tau=1,\ldots,T$. $V_x^0=e^{-r T}\max(K-e^x,0)$ $\forall x=1,\dots,N+1$ with
$i=0=M1$ and $i=N+1=M2$.\\
Note that the solutions to the differential equation must satisfy the boundary
condition $V_x^{\tau}\geq e^{-\tau}\max(K-e^x,0)$  $\forall
x=1,\dots,N+1$.\\
In order to get the option price of the American option we need to use the
`Bellman principle of dynamics programming'
\begin{eqnarray*}
V(S_t)&=&\max(f(S_t), PV_t[V(S_t+ds, t+dt)])\\
V_x^{\tau}&=&\max\left(e^{\tau}\max(K-e^x,0),V_{x+1}^{\tau+1}\right)
\end{eqnarray*}
in which $V_x^{\tau}$ and $V_{x+1}^{\tau+1}$ are calculated either with the FTCS or the Crank-Nicolson scheme. We did'n't actually calculate values of the American Option because this was too time consuming. Alhough this method shows us that for calculating the price of an American Option Finite Difference is highly preferable over the Monte Carlo Method.\\



